Suppose I have a nerve fiber consisting of several axons all running in parallel to each other. When an action potential is generated in a certain axon, this will alter the concentration of sodium and potassium ions inside the axon, and in the extracellular fluid around that axon.

The question is, will this change in ion concentration in the extracellular fluid, alter the threshold potential of an adjacent axon, which shares the same extracellular fluid?

1 Answer
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Yes, activity in other cells can affect the surrounding electric field and thereby influence the behavior of nearby neurons. This phenomenon, known as ephatic coupling, can be important but is usually small.

For many cases, it is likely too small to be of physiological importance. Note that individual action potentials are quite transient and, especially in myelinated axons, the resulting currents are local and small. Therefore it's unlikely that a single neuron's activity will have a noticeable effect on nearby neurons.

However, in unmyelinated axon tracts and/or when a larger group of neurons is active, ephatic coupling is more likely. In particular, it seems to play a role in synchronizing cell populations during oscillations. For example, Anastassiou et al (2011) found that firing of cortical neurons could be entrained by the extracellular field and affected the intracellular membrane potential by about 0.5 mV.

Edit:
For your particular case, action potential propagation in one axon is very unlikely to affect the other axons in any relevant way in a tract of myelinated axons. This is especially true because the initiation of action potentials does not occur along the axon itself, but where the axon is just branching from the cell body (the axon initial segment). Threshold is important at the initiation site, but is not really relevant in the tract itself, where the axon is just conducting the signal it receives from the initiation site.